As mentioned in [1]}, this direction of research is indeed as old as the field of parameterized algorithms.
The result of Bodlaender [2]}
on on the problem for computing a treewidth of a graph
belongs to this class of results. Also, the \(f(k)\cdot n^3\) -time and \(h(k)\cdot n^2\) -time algorithms for the Disjoint Paths problem
by Roberson and Seymour [3]} and by Kawarabayashi et al. [4]}, respectively,
belong to this class of results.
There are more recent results in this direction, for instance, Odd Cycle Transversal [5]}, [6]},
Subgraph Isomorphism [7]}, Planarization [8]}, and Treewidth [9]}.
